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INTRODUCTION to SPECTRAL SEQUENCES 1. Introduction in Homological Algebra and Algebraic Topology, a Spectral Sequence Is a Means

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INTRODUCTION to SPECTRAL SEQUENCES 1. Introduction in Homological Algebra and Algebraic Topology, a Spectral Sequence Is a Means INTRODUCTION TO SPECTRAL SEQUENCES HUAN VO 1. Introduction In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Roughly speaking, a spectral sequence is a r r sequence (E ; d )r≥1 of bicomplexes, which is can think of as a \book" with infinitely many \pages". Every time we go to the next page, we get a bit closer to the homology of the complex we want to compute. Spectral sequences were invented in the 1940s, independently, by J. Leray and R. C. Lyndon. Leray was a German prisoner of war from 1940 through 1945, during World War II. This notes aims to give a brief introduction to spectral sequences from a purely homological algebra point of view. Specifically, we only describe the set-up of spectral sequences, without going into their many exciting applications in topology and geometry. For this notes, we closely follow the treatment found in [1]. We only assume very basic homological algebra facts. In particular, we assume the readers are comfortable with the notions of complexes, homology, exact sequence in homologies. Some technical proofs which are not very essential to understanding the flow of ideas are referenced in [1]. The organization of the notes goes as follows. In Section 2, we introduce our objects of study: bicomplexes and their total complexes. Our main goal is to compute the homology of the total complex of a complex. In Section 3, we describe the essential concept of a filtration of a complex and how to go from a filtration of a complex to an exact couple, which then gives rise to spectral sequence, by a recursive process. In Section 4, we discuss spectral sequences and show that a spectral sequence arising from a filtration of a complex \converges" to the homology of the complex. Here by convergence, we mean only up to extension, i.e. there exists a short exact sequence p−1 p 1 0 −! Φ Hn −! Φ Hn −! Ep;q −! 0: In Section 5, we apply our general discussion to the case of the total complex of a bicomplex and describe a concrete way to compute the first two pages of the spectral sequence using iterated homologies. Finally, in Section 6, we illustrate how to use the machinery we have developed to show that the definition of the Tor functor is independent of the variable resolved. In this notes R will denote a commutative ring with 1. It follows that left R-modules are the same as right R-modules, and we just call them R-modules. 2. Bicomplexes Definition 2.1. A graded module is an indexed family M = (Mp)p2Z of R-modules. Definition 2.2. Let M, N be graded modules and let a 2 Z .A graded map of degree a, denoted by f : M ! N, is a family of R-module homomorphisms f = (fp : Mp ! Np+a)p2Z . The degree of f is a, and we denote it by deg(f) = a. Let M and M 0 be graded modules, then M 0 is a submodule of M, denoted M 0 ⊆ M, if each 0 0 Mp is a submodule of Mp for all p 2 Z . If M is a submodule of M, then we can form the 0 0 quotient M=M = (Mp=Mp)p2Z . If f : M ! N is a graded map of degree a, then the kernel 1 INTRODUCTION TO SPECTRAL SEQUENCES 2 of f is the graded module ker f = (ker fp)p2Z ⊆ M and the image of f is the graded module im f = (im fp−a)p2Z ⊆ N. A sequence of graded maps f g A −! B −! C is exact if im f = ker g, i.e. if im fp−a = ker gp for all p 2 Z . Definition 2.3. A complex (or chain complex) is an ordered pair (C; d), where C is a graded module and d is a graded map of degree −1 that satisfies dd = 0. The map d is called the differential. dn+2 dn+1 dn dn−1 C : ··· −−−! Cn+1 −−−! Cn −! Cn−1 −−−!· · · : The condition dd = 0 implies that im dn+1 ⊆ ker dn for all n 2 Z . The nth homology module of (C; d) is defined as Hn(C) = ker dn=im dn+1: Definition 2.4. A bigraded module is a doubly indexed family M = (Mp;q)(p;q)2Z ×Z of R- modules. Definition 2.5. Let M, N be bigraded modules and let (a; b) 2 Z × Z .A bigraded map of degree (a; b), denoted by f : M ! N, is a family of R-module homomorphisms f = (fp;q : Mp;q ! Np+a;q+b)(p;q)2Z ×Z . The bidegree of f is (a; b), and we denote it by deg(f) = (a; b). Let M and M 0 be bigraded modules, then M 0 is a submodule of M, denoted M 0 ⊆ M, if each 0 0 Mp;q is a submodule of Mp;q for all (p; q) 2 Z × Z . If M is a submodule of M, then we can form 0 0 the quotient M=M = (Mp;q=Mp;q)(p;q)2Z ×Z . If f : M ! N is a bigraded map of bidegree (a; b), then the kernel of f is the bigraded module ker f = (ker fp;q)(p;q)2Z ×Z ⊆ M and the image of f is the bigraded module im f = (im fp−a;q−b)(p;q)2Z ×Z ⊆ N. A sequence of bigraded maps f g A −! B −! C is exact if im f = ker g, i.e. if im fp−a;q−b = ker gp;q for all (p; q) 2 Z × Z . Definition 2.6. A differential bigraded module is an ordered pair (M; d), where M is a bigraded module and d: M ! M is a bigraded map with dd = 0. The map d is called the differential. If (M; d) is a bigraded module, where d has bidegree (a; b), then its homology H(M; d) is the bigraded module whose (p; q) term is ker dp;q H(M; d)p;q = : im dp−a;q−b Definition 2.7. A bicomplex (or double complex) is an ordered triple (M; d0; d00), where M = 0 00 (Mp;q) is a bigraded module, d ; d : M ! M are bigraded maps of bidegree (−1; 0) and (0; −1), respectively that satisfy (i) d0d0 = 0, (ii) d00d00 = 0, 0 00 00 0 (iii) dp;q−1dp;q + dp−1;qdp;q = 0 0 00 Thus we can visualize the bicomplex (M; d ; d ) as follows. Put Mp;q at the lattice point (p; q) 0 00 of Z × Z . The map d points to the left and the map d points down. The first condition says that each row forms a complex with differentials d0 and the second condition says that each column forms a complex with differentials d00. The third condition says that each square of the lattice anticommutes. INTRODUCTION TO SPECTRAL SEQUENCES 3 Definition 2.8. If (M; d0; d00) is a bicomplex, then its total complex, denoted by Tot(M), is the complex with nth term M Tot(M)n = Mp;q p+q=n P 0 00 and with differentials Dn : Tot(M)n ! Tot(M)n−1 given by Dn = p+q=n(dp;q + dp;q). 0 dp−1;q+1 Mp−2;q+1 Mp−1;q+1 00 dp−1;q+1 0 dp;q Mp−1;q Mp;q 00 dp;q Mp;q−1 Lemma 2.9. If (M; d0; d00) is a bicomplex, then (Tot(M);D) is a complex. Proof. We need to check that indeed Dn : Tot(M)n ! Tot(M)n−1 and Dn−1Dn = 0. This is where condition (iii) in definition 2.7 comes into play. The readers are encouraged to draw the picture and see that directly. 3. Filtrations and Exact Couples Definition 3.1. A filtration of an R-module M is a family (Mp)p2Z of submodules of M such that · · · ⊆ Mp−1 ⊆ Mp ⊆ Mp+1 ⊆ · · · : For each p 2 Z , the module Mp=Mp−1 is called a factor module of the filtration. p Definition 3.2. A filtration of a complex (C; d) is a sequence of graded submodules (F C)p2Z of C such that · · · ⊆ F p−1C ⊆ F pC ⊆ F p+1C ⊆ · · · that satisfies p p d:(F C)n ! (F C)n−1 p p for all n; p. To simplify notation, we denote (F C)n by Fn . Note that the following diagram commutes. p−1 p p+1 ··· Fn+1 Fn+1 Fn+1 ··· p−1 p p+1 ··· Fn Fn Fn ··· p−1 p p+1 ··· Fn−1 Fn−1 Fn−1 ··· . p The horizontal maps are inclusions and the vertical maps are restrictions of d to Fn . The nth row form a filtration of Cn. INTRODUCTION TO SPECTRAL SEQUENCES 4 Definition 3.3. Let (M; d0; d00) be a bicomplex. The first filtration (or vertical filtration) of Tot(M) is given by I p M ( F Tot(M))n = Mi;n−i i≤p = · · · ⊕ Mp−2;q+2 ⊕ Mp−1;q+1 ⊕ Mp;q: Here we use the convention that n = p + q. Thus we have I p−1 I p I p+1 · · · ⊆ ( F Tot(M))n ⊆ ( F Tot(M))n ⊆ ( F Tot(M))n ⊆ · · · I p a filtration of Tot(M)n. Pictorially, ( F Tot(M))n is the direct sum of the terms on the line x + y = n which lie to the left of the vertical line x = p.
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